Content deleted Content added
Erel Segal (talk | contribs) |
|||
| (47 intermediate revisions by 26 users not shown) | |||
Line 1:
{{Short description|Attribute of machine learning models}}
{{Machine learning bar}}
The '''sample complexity''' of a [[machine learning]] algorithm represents the number of training-samples that it needs in order to successfully learn a target function.
Line 8 ⟶ 9:
* The strong variant takes the worst-case sample complexity over all input-output distributions.
The [[No
However, if we are only interested in a particular class of target functions (e.g., only linear functions) then the sample complexity is finite, and it depends linearly on the [[VC dimension]] on the class of target functions.<ref name=":0" />
==Definition==
Let <math>X</math> be a space which we call the input space, and <math>Y</math> be a space which we call the output space, and let <math>Z</math> denote the product <math>X\times Y</math>. For example, in the setting of binary classification, <math>X</math> is typically a finite-dimensional vector space and <math>Y</math> is the set <math>\{-1,1\}</math>.
Fix a hypothesis space <math>\mathcal H</math> of functions <math>h\colon X\to Y</math>. A learning algorithm over <math>\mathcal H</math> is a computable map from <math>Z
Fix a loss function <math>
:<math>\mathcal E(h) :=\mathbb E_\rho[
In our setting, we have <math>h=
\mathcal E^*_\mathcal{H} = \underset{h \in \mathcal H}{\inf}\mathcal E(h).
</math>Set <math>h_n=\mathcal{A}(S_n)</math>, for each [[sample size]] <math>n</math>. <math>h_n</math> is a [[random variable]] and depends on the random variable <math>S_n</math>, which is drawn from the distribution <math>\rho^n</math>. The algorithm <math>\mathcal{A}</math> is called '''consistent''' if <math> \mathcal E(h_n) </math> probabilistically converges to <math> \mathcal E_\mathcal H^*</math>. In other words, for all <math>\epsilon, \delta > 0</math>, there exists a positive integer <math>N</math>, such that, for all sample sizes <math>n \geq N</math>, we have
<math display="block">
\Pr_{\rho^n}[\mathcal E(h_n) - \mathcal E^*_\mathcal{H}\geq\varepsilon]<\delta.
</math>The '''sample complexity''' of <math>
In others words, the sample complexity <math>N(\rho,\epsilon,\delta)</math> defines the rate of consistency of the algorithm: given a desired accuracy
In [[Probably approximately correct learning|
==Unrestricted hypothesis space: infinite sample complexity==
<span id=
One can ask whether there exists a learning algorithm so that the sample complexity is finite in the strong sense, that is, there is a bound on the number of samples needed so that the algorithm can learn any distribution over the input-output space with a specified target error. More formally, one asks whether there exists a learning algorithm <math>
<math display="block">
\sup_\rho\left(\Pr_{\rho^n}[\mathcal E(h_n) - \mathcal E^*_\mathcal{H}\geq\varepsilon]\right)<\delta,
</math>
\sup_\rho\left(\Pr_{\rho^n}[\mathcal E(f_n) - \mathcal E^*_\mathcal{H}\geq\varepsilon]\right),▼
</math>one must either▼
Thus, in order to make statements about the rate of convergence of the quantity
<math display="block">
</math>
*constrain the space of probability distributions <math>\rho</math>, e.g. via a parametric approach, or
*constrain the space of hypotheses <math>\mathcal H</math>, as in distribution-free approaches.
==Restricted hypothesis space: finite sample-complexity==
The latter approach leads to concepts such as [[VC dimension]] and [[Rademacher complexity]] which control the complexity of the space <math>\mathcal H</math>. A smaller hypothesis space introduces more bias into the inference process, meaning that <math>\mathcal E^*_\mathcal{H}</math> may be greater than the best possible risk in a larger space. However, by restricting the complexity of the hypothesis space it becomes possible for an algorithm to produce more uniformly consistent functions. This trade-off leads to the concept of [[regularization (mathematics)|regularization]].<ref name = "Rosasco" />
Line 52 ⟶ 58:
# The VC dimension of <math>\mathcal H</math> is finite.
# <math>\mathcal H</math> is a uniform [[Glivenko-Cantelli class]].
This gives a
=== An example of a PAC-learnable hypothesis space ===▼
=== Sample-complexity bounds ===
<span id='bounds'></span>
Suppose <math>\mathcal H</math> is a class of binary functions (functions to <math>\{0,1\}</math>). Then, <math>\mathcal H</math> is <math>(\epsilon,\delta)</math>-PAC-learnable with a sample of size:
<ref>{{Cite journal|title=The optimal sample complexity of PAC learning
|journal=J. Mach. Learn. Res.|volume=17|issue=1|pages=1319–1333|author=Steve Hanneke|year=2016|arxiv=1507.00473|url=https://www.jmlr.org/papers/v17/15-389.html}}</ref>
<math display="block">
N = O\bigg(\frac{VC(\mathcal H) + \ln{1\over \delta}}{\epsilon}\bigg)
</math>
where <math>VC(\mathcal H)</math> is the [[VC dimension]] of <math>\mathcal H</math>.
Moreover, any <math>(\epsilon,\delta)</math>-PAC-learning algorithm for <math>\mathcal H</math> must have sample-complexity:<ref>{{Cite journal|doi=10.1016/0890-5401(89)90002-3|title=A general lower bound on the number of examples needed for learning|journal=Information and Computation|volume=82|issue=3|pages=247|year=1989|last1=Ehrenfeucht|first1=Andrzej|last2=Haussler|first2=David|last3=Kearns|first3=Michael|last4=Valiant|first4=Leslie|doi-access=free}}</ref>
<math display="block">
N = \Omega\bigg(\frac{VC(\mathcal H) + \ln{1\over \delta}}{\epsilon}\bigg)
</math>
Thus, the sample-complexity is a linear function of the [[VC dimension]] of the hypothesis space.
Suppose <math>\mathcal H</math> is a class of real-valued functions with range in <math>[0,T]</math>. Then, <math>\mathcal H</math> is <math>(\epsilon,\delta)</math>-PAC-learnable with a sample of size:
<ref name=mr15>{{cite book|first1=Martin|last1=Anthony|first2=Peter L.|last2=Bartlett|title=Neural Network Learning: Theoretical Foundations|year=2009|isbn=9780521118620}}</ref><ref>{{cite conference|title=On the Pseudo-Dimension of Nearly Optimal Auctions|year=2015|conference=NIPS|url=http://papers.nips.cc/paper/5766-on-the-pseudo-dimension-of-nearly-optimal-auctions|arxiv=1506.03684|last1=Morgenstern|first1=Jamie|author1-link=Jamie Morgenstern|last2=Roughgarden|first2=Tim|pages=136–144|publisher=Curran Associates}}</ref>
<math display="block">
N = O\bigg(T^2\frac{PD(\mathcal H)\ln{T\over \epsilon} + \ln{1\over \delta}}{\epsilon^2}\bigg)
</math>
where <math>PD(\mathcal H)</math> is [[VC dimension#Generalizations|Pollard's pseudo-dimension]] of <math>\mathcal H</math>.
==Other settings==
In addition to the supervised learning setting, sample complexity is relevant to [[semi-supervised learning]] problems including [[Active learning (machine learning)|active learning]],<ref name="Balcan">{{cite journal |doi = 10.1007/s10994-010-5174-y|title = The true sample complexity of active learning|journal = Machine Learning|date = 2010|volume = 80|issue =
==Efficiency in robotics==
▲== An example of a PAC-learnable hypothesis space ==
A high sample complexity means that many calculations are needed for running a [[Monte Carlo tree search]].<ref>{{cite conference |title=Monte-carlo tree search by best arm identification |author=Kaufmann, Emilie and Koolen, Wouter M |conference=Advances in Neural Information Processing Systems |pages=4897–4906 |year=2017 }}</ref> It is equivalent to a [[Model-free (reinforcement learning)|model-free]] brute force search in the state space. In contrast, a high-efficiency algorithm has a low sample complexity.<ref>{{cite conference |title=The chin pinch: A case study in skill learning on a legged robot |author=Fidelman, Peggy and Stone, Peter |conference=Robot Soccer World Cup |pages=59–71 |year=2006 |publisher=Springer }}</ref> Possible techniques for reducing the sample complexity are [[metric learning]]<ref>{{cite conference |title=Sample complexity of learning mahalanobis distance metrics |author=Verma, Nakul and Branson, Kristin |conference=Advances in neural information processing systems |pages=2584–2592 |year=2015 }}</ref> and model-based reinforcement learning.<ref>{{cite arXiv |title=Model-ensemble trust-region policy optimization |author=Kurutach, Thanard and Clavera, Ignasi and Duan, Yan and Tamar, Aviv and Abbeel, Pieter |eprint=1802.10592 |year=2018 |class=cs.LG }}</ref>
▲Let ''X'' = '''R'''<sup>''d''</sup>, ''Y'' = {-1, 1}, and let <math>\mathcal H</math> be the space of affine functions on ''X'', that is, functions of the form <math>x\mapsto \langle w,x\rangle+b</math> for some <math>w\in\R^d,b\in\R</math>. This is the linear classification with offset learning problem. Now, note that four coplanar points in a square cannot be shattered by any affine function, since no affine function can be positive on two diagonally opposite vertices and negative on the remaining two. Thus, the VC dimension of <math>\mathcal H</math> is 3, in particular finite. It follows by the above characterization of PAC-learnable classes that <math>\mathcal H</math> is PAC-learnable, and by extension, learnable.
==
* [[Active learning (machine learning)]]
▲In addition to the supervised learning setting, sample complexity is relevant to [[semi-supervised learning]] problems including [[active learning]],<ref name="Balcan">{{cite journal |doi = 10.1007/s10994-010-5174-y|title = The true sample complexity of active learning|journal = Machine Learning|date = 2010|volume = 80|issue = 2-3|pages = 111–139|first = Maria-Florina|last = Balcan}}</ref> where the algorithm can ask for labels to specifically chosen inputs in order to reduce the cost of obtaining many labels. The concept of sample complexity also shows up in [[reinforcement learning]],<ref>{{citation |last = Kakade | first = Sham | title = On the Sample Complexity of Reinforcement Learning | place = University College London | publisher = Gatsby Computational Neuroscience Unit. | series = PhD Thesis | year = 2003 | url = http://www.ias.tu-darmstadt.de/uploads/Research/NIPS2006/SK.pdf}}</ref> [[online machine learning|online learning]], and unsupervised algorithms, e.g. for [[dictionary learning]].<ref>{{cite journal |last1 = Vainsencher | first1 = Daniel | last2 = Mannor | first2 = Shie | last3 = Bruckstein | first3 = Alfred | title = The Sample Complexity of Dictionary Learning | journal = Journal of Machine Learning Research | volume = 12 | pages = 3259–3281 | date = 2011 | url = http://www.jmlr.org/papers/volume12/vainsencher11a/vainsencher11a.pdf}}</ref>
==References==
| |||